I'm not a physicist, and so I realize this contradiction is likely my layman misunderstanding. I appreciate your time in teaching me.
Dr. Cox seems to describe, I think, the Higgs Boson as though it were just another gauge boson, i.e. one of the fundamental "force carrier" particles by which force is transmitted to other particles. Except in this case, the H.B. is not transmitting force but imparting mass. He goes on to say explicitly that Higgs Bosons exist in vast numbers all around us and are constantly bombarding all other particles at all times. It is through this bombardment that mass is gained.
Is my summary and interpretation correct? If so, is his explanation accurate? My understanding was the H.B. is more or less a byproduct of the Higgs field itself, the result of its excitation, and it is interaction with the field itself, not the H.B., that endows other particles with mass. This slide from the Berkeley lecture lists the H.B. as neither a gauge boson particle, nor a matter particle, but an entirely different, third type of particle.
Does the Higgs field exist because there's an abundance of H.B. particles just moving around, themselves giving rise to the Higgs field? Or does the Higgs field simply exist everywhere, and by virtue of its existence, occasionally an H.B. pops out? (Of this last point, I am certain. Several sources have directly stated that what makes the Higgs field unique is that its default energy state is nonzero, i.e. unlike the "electron field", it exists everywhere, even in a vacuum).
Do Higgs Bosons happen on their own in nature all the time? Rarely? Or do they only happen when the field is excited under deliberate conditions, e.g. within a man-made particle accelerator? Or is my question malformed and running afoul of the wave/particle duality?
A final summary of my running understanding:
Consider an electron and a positron, some distance apart. They each experience an attractive force to one another, via the electromagnetic field they produce. The actual means by which this force is transmitted, the means by which the electron "knows about" the positron, is by way of a gauge boson, in this case the photon. "The photon is the mediating particle of the EM field". You quite literally have photons moving back and forth between the electron and positron. They serve as the actual means of transmission of force/energy/momentum transfer and exchange between the two particles. (I'm referencing some illustration I saw in a video that quite literally had an animation of photons bouncing back and forth from, in their example, nuclear protons and their orbiting electrons).
I think I'm just at a loss to explain the relationship between and difference among the Higgs field and the H.B. itself. Explanations in lectures and the media seem to vary from "With respect to endowing mass, the H.B. does it all" to "The Higgs field does it all; the H.B. is just a theoretical consequence of the existence of the Higgs field; finding the H.B. is strong confirmation that the Higgs field itself exists".